Remarks on W1, p-quasiconvexity, interpenetration of matter, and function spaces for elasticity

R. D. James, S. J. Spector

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We show that, under mild hypotheses on the elastic energy function, the minimizer of the energy in the space {f∈W1,p:det∇f>0,1≦p<n} of a nonlinear elastic ball subject to the severe compressive boundary conditions f(x)=λx,λ≪1 will not be the expected uniform compression f (x) = λx. To show this, we construct competitors in this space that reduce the energy but interpenetrate matter. We also prove that the W1, p-quasiconvexity condition of Ball and Murat [1984] is a necessary condition for a local minimum in a setting that includes nonlinear elasticity. This theorem is well suited to analyses of the formation of voids in nonlinear elastic materials. Our analysis illustrates the delicacy of the choice of function space for nonlinear elasticity.

Original languageEnglish (US)
Pages (from-to)263-280
Number of pages18
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Issue number3
StatePublished - May 1 1992

Bibliographical note

Funding Information:
R. D. J. acknowledges the support of the National Science Foundation and the Air Force Office of Scientific Research through NSF/DMS-8718881. S. J. S. acknowledges the support the Institute for Mathematics and its Applications, the National Science Foundation through NSF/DMS-8810653, and the Air Force Office of Scientific Research through AFOSR-880200.


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